Question

If the problem type is labeled “ClickAndPlace”, click the item to select it and then click the location where it belongs; otherwise, click the item and drag it into place.

Express answer in exact form.

Find the area of the smaller segment whose chord is 8" long in a circle with an 8" radius.


help me i do not have a diagram

Answer 1

Draw a circle with center at point "O" and 8" radius.
Draw an 8" chord on the circle ,
and call the points the chord touches the circle "A" and "B".
Note that if you consider the points "A","O","B" as vertices of a triangle - it is an equilateral triangle (all sides are 8") , so all it's angles are 60 degrees.


Now see this link:
http://www.ajdesigner.com/phpcircle/circle_segment_area_k.php

We need to find K.

\[K=\frac{r^2(\theta-\sin \theta)}{2}\]

In our case, r=8 and
\[\theta = 60 degrees = \frac{\pi}{3} radians\]

So,
\[K=\frac{64}{2}\left( \frac{\pi}{3}-\frac{\sqrt{3}}{2} \right)=\frac{32\pi}{3}-16\sqrt{3}\]